Tension

Chang Xu (Fall 2017)

This topic covers Tension.

Main Idea

What is Tension?

Tension is the force exerted by a rope (or anything that can be used to hang another object) on the object that is hanging from it. Usually, it is ropes and cables that have the tension force. In general, anything that is flexible can pull an object and have tension force. In consequence, the tension force can only be a pulling force. The rope will eventually go slack if someone tries to push with a rope, and it will act like just an object. Later we will see that this concept will help with draing force diagrams with the force of tension always pulling the object.

Tension is considered a contact force which means that the force is exerted when objects are touching. Usually, the force of tension is the force that is transmitted through a rope. If someone is pulling on a block with a rope, the person exerting force on the rope which transmits that force to the block. In problems, the ropes and cables will usually be massless, which perfectly transfers the force.

Mathematical Modeling

When calculating for tension force, tension force should be treated as one of the forces that is in the problem. The most important formula that will be used in this will be:

Fnet = Mass x Acceleration

This is based on Newton's Second Law, which applies because there will be multiple forces acting on the object so the forces must be added together. Because of this, we will also have to add all the forces together to find Fnet:

Fnet = F1 + F2 + ... + Fn for n number of forces.

A lot of times for simple force of tension problems, the acceleration may be 0, which causes our Fnet to be 0 (because Fnet = Mass x Acceleration). Acknowledging this will make calculating forces easier. This case will have to do with Newton's Third Law which sates that every force has an equal and opposite force; therefore, Fnet will be 0.

When acceleration is not 0, we will have to solve for Fnet and then for Ft (force of tension).

How To Calculate Tension Force

Because most situations of tension are different, there is no set way of calculating the force of tension.

Example Problems

Example 1: Angled rope pulling on a box

A 2.0kg box of toy box is being pulled across a table by a rope at an angle θ=60º as seen below (ignore friction). The tension in the rope causes the box to slide across the table to the right with an acceleration of 3.0 m/s^2

What is the tension in the rope?

First draw a force diagram of all the forces acting on the box.

Now use Newton's second law. The tension is directed both vertically and horizontally, so it's a little unclear which direction to choose. However, since the acceleration is going horizontally, and since the tension is the only force directed horizontally, use Newton's second law in the horizontal direction.

a= ΣF/m (use Newtons's second law for the horizontal direction)
3.0 m/s^2=Tcos60º/2.0kg (plug in the horizontal acceleration, mass, and horizontal forces)
Tcos60º=(3.0 m/s^2)(2.0kg)
T=[(3.0 m/s^2)(2.0kg)]/(cos60º)
T=12N

Example 2: Box hanging from two ropes

A 0.25 kg container hangs at rest from two strings secured to the ceiling and wall respectively. The diagonal rope under tension T1 is directed at an angle θ=30º from the horizontal direction as seen below.

What are the tensions (T1 and T2) in the two strings?

First draw a force diagram of all the forces acting on the container.

Now use Newton's second law. There are tensions directed both vertically and horizontally, so again it's a little unclear which direction to choose. However, since there is force of gravity (a vertical force), start with Newton's second law in the vertical direction.

a=ΣF/m (use Newton's second law for the vertical direction)
0=(T2*sin30º-Fg)/0.25kg
T2=Fg/(sin30º)
T2=mg/(sin30º)
T2=[(0.25kg)(9.8m/s²)]/(sin30º)
T2=4.9N

Now that we know T2 we can solve for the tension T1 using Newton's second law for the horizontal direction.

a=ΣF/m (use Newton's second law for the horizontal direction)
0=(T2*cos30º-T1)/0.25kg (plug in the horizontal acceleration, mass, and horizontal forces)
T1=T2*cos30º
T1=(4.9N)*cos30º
T1=4.2N

Connectedness

How is this topic connected to something that you are interested in?

When I was little, I went to a department store and I saw a transparent elevator. I never knew how elevator operated but through the transparent elevator, I realized that the pulling force of the ropes was what was keeping the elevator moving. This pulling force is tension and I later realized that this tension force is evident everywhere in my daily life.

How is it connected to your major?

I am not exactly sure how tension is connected to industrial engineering, which is my major. However, I can say that tension is a very basic concept in physics related to force and it is important to understand physics mechanism in studying industrial engineering.

Is there an interesting industrial application?

Tension force can be seen in everyday life, just like the elevator example I mentioned above. Tension force is applied when I pull a clothing tags.

Tension

Contents

Main Idea

What is Tension?

Mathematical Modeling

How To Calculate Tension Force

Example Problems

Example 1: Angled rope pulling on a box

Example 2: Box hanging from two ropes

Connectedness

See also

Further reading

External links

References

Navigation menu